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Should Major League Baseball stadiums be standardized? Home runs are an exciting part of professional baseball. However, not all stadiums are created equally; some are harder to hit home runs in than others, which can have a big effect on pitchers and batters.

In this lesson, students use a quadratic function to model the trajectory of the average professional home run and debate whether Major League Baseball stadiums should all be designed the same.

REAL WORLD TAKEAWAYS

MLB stadiums are not standardized. The outfield walls have different heights and distances from home plate.

MATH OBJECTIVES

Write and solve quadratic equations in a real-world context

Evaluate a quadratic function at given values

Sketch a quadratic function from an equation written in vertex form, and interpret the key features

This complex task is best as a culminating unit activity after students have developed formal knowledge and conceptual understanding.

Algebra 1

Quadratics & Solving

Algebra 1

Quadratics & Solving

Content Standards
A.SSE.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (a) Factor a quadratic expression to reveal the zeros of the function it defines. (b) Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (c) Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15<sup>t</sup> can be rewritten as (1.15<sup>1/12</sup>)<sup>12t</sup> ≈ 1.012<sup>12t</sup> to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
A.CED.1
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
A.REI.1
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A.REI.4
Solve quadratic equations in one variable. (a) Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x — p)<sup>2</sup> = q that has the same solutions. Derive the quadratic formula from this form. (b) Solve quadratic equations by inspection (e.g., for x<sup>2</sup> = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
F.IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (a) Graph linear and quadratic functions and show intercepts, maxima, and minima. (b) Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (c) Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. (d) (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (e) Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Mathematical Practices
MP.1
Make sense of problems and persevere in solving them.
MP.2
Reason abstractly and quantitatively.
MP.3
Construct viable arguments and critique the reasoning of others.
MP.4
Model with mathematics.
MP.5
Use appropriate tools strategically.
MP.8
Look for and express regularity in repeated reasoning.